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In this paper, we study the dynamics of a vertically emitting micro-cavity operated in the Gires–Tournois regime that contains a semiconductor quantum-well and that is subjected to strong time-delayed optical feedback and detuned optical injection. Using a first principle time-delay model for the optical response, we disclose sets of multistable dark and bright temporal localized states coexisting on their respective bistable homogeneous backgrounds. In the case of anti-resonant optical feedback, we identify square-waves with a period of twice the round-trip in the external cavity. Finally, we perform a multiple time scale analysis in the good cavity limit. The resulting normal form is in good agreement with the original time-delayed model.

To understand the non-equilibrium relaxation dynamics of a liquid droplet on a switchable substrate the interplay of different length- and time-scales needs to be understood. We present a method to map the microscopic information, resulting from a molecular dynamics simulation, to a mesoscopic scale, reflected by a thin film model. After a discussion of the mapping procedure we first analyze the relaxation of a liquid droplet upon switching the wettability of the substrate. Further, we show that a nearly identical mapping procedure can be used for the description of two coalescing droplets. With our procedure we take a first step to extend the mapping from the equilibrium case to non-equilibrium wetting dynamics, thus allowing for a quantitative multi-scale analysis., 34 pages, 15 figures

We study theoretically the mechanisms of square-wave (SW) formation in vertical external-cavity Kerr-Gires-Tournois interferometers in presence of anti-resonant injection. We provide simple analytical approximations for their plateau intensities and for the conditions of their emergence. We demonstrate that SWs may appear via a homoclinic snaking scenario, leading to the formation of complex-shaped multistable SW solutions. The resulting SWs can host localized structures and robust bound-states., 4 pages, 5 figures, submitted to optics letters

The development of substrates with a switchable wettability is on a fast pace. The limit of switching frequencies and contact angle differences between substrate states are steadily pushed further. We investigate the behavior of a droplet on a homogeneous substrate, which is switched between two wettabilities for a large range of switching frequencies. Here, we are particularly interested in the dependence of the wetting behavior on the switching frequency. We show, that results obtained on the particle level via molecular dynamics simulations and on the continuum level via the thin-film model are consistent. Predictions of simple models as the molecular theory of wetting (MKT) and analytical calculations based on the MKT also show good agreement and offer deeper insights into the underlying mechanisms.

We investigate the destabilization mechanisms of dissipative solitons in inhomogeneous nonlinear res-onators subjected to injection and to time-delayed feedback. We consider the paradigmatic Lugiato-Lefever model describing inhomogeneous driven nonlinear optical resonator. We analyze the pinning-depinning transition of dissipative solitons by introducing a potential induced by the inhomogeneity. Fur-ther, we identify conditions under which these structures are destabilized and describe different bifurca-tion scenarios. We show that the combined influence of inhomogeneities and delayed feedback induces an Andronov-Hopf-bifurcation that leads to oscillations of the dissipative soliton around the inhomogene-ity. Finally, we show that for large values of the feedback strength, the dissipative solitons escapes from the potential well and starts to drift. (c) 2021 Elsevier Ltd. All rights reserved.

Degenerate laser cavity have been used for second-harmonic generation [1] , [2] and spatially incoherent laser emission using solid-state gain medium and monolithic VCSEL with large area of excitation [3] , [4] . In this work we investigate cw spatial coherent states of a degenerate laser cavity using specially design 1/2-VCSEL based on III-V semiconductor technology, and high numerical aperture optical system, in order to be used in spatio-temporal localization of light also know as Light Bullets [5] . The external cavities implemented are made up two(four) lenses in 4-(-8)f self-imaging configurations. A Zeemax and optical physics study was carried out to design the optimized system of lenses to ensure high numerical aperture and high resolution down to wavelength scale with minimal aberrations for off axis operation. The Semiconductor Gain Mirror (1/2-VCSEL) have been designed based on GaAs materials at 1060nm wavelength and for efficient optical pumping at 800nm using commercial diode laser. The high reflectivity mirror is designed with an GaAs/AlAs bragg mirror and the optical gain is provided through a multi-quantum-well strained-balanced InGaAs/GaAsP active region. The 1/2-VCSEL was designed with the intent to minimize all waveguiding effect such as the thermal lens and strong self-guiding through gain saturation.

We present a generalization of the Haus master equation in which a dynamical boundary condition allows to describe complex pulse trains such as the Q-switched and harmonic transitions of passive mode-locking as well as the weak interactions between localized states. As an example, we investigate the influence of group velocity dispersion on the stability boundaries of the Q-switched regime. We compare our results with that of a time-delayed system., Comment: 4 pages, 5 figures

We analyze both theoretically and experimentally the impact of optical feedback on the dynamics of external-cavity mode-locked semiconductor lasers (VECSELs) operated in the long cavity regime [1]. In particular, by choosing certain ratios between the cavity round-trip time and the feedback delay, we show that feedback acts as a solution discriminator that either reinforces or hinders the appearance of one of the multiple harmonic arrangements of temporal localized structures. For the theoretical modeling, the delayed differential equation model [2] is extended by a term describing the optical feedback. Denoting by A the amplitude of the optical field, G the gain, and Q the saturable losses, the model reads\begin{align*} & \frac{{\dot A}}{\gamma } = \sqrt \kappa \exp \left[ {\frac{{1 - i{\alpha _g}}}{2}G\left( {t - {\tau _c}} \right) - \frac{{1 - i{\alpha _a}}}{2}Q\left( {t - {\tau _c}} \right)} \right]A\left( {t - {\tau _c}} \right) - A(t) + \eta {e^{i{{\Omega }}}}A\left( {t - {\tau _f}} \right),\tag{1} \\ & \dot G = {g_0} - {{\Gamma }}G - {e^{ - Q}}\left( {{e^G} - 1} \right)|A{|^2},\quad \dot Q = {q_0} - Q - s\left( {1 - {e^{ - Q}}} \right)|A{|^2}\tag{2}\end{align*}where time has been normalized to the SA recovery time, α g and α a are the linewidth enhancement factors of the gain and absorber sections, respectively, κ the fraction of the power remaining in the cavity after each round- trip, g 0 the pumping rate, Γ the gain recovery rate, q 0 is the value of the unsaturated losses which determines the modulation depth of the SA, τ c the round-trip time in the cavity, s the ratio of the saturation energy of the SA and of the gain sections and γ is the bandwidth of the spectral filter, η is the feedback rate, Ω is the feedback phase and τ f the round-trip time in the feedback loop. For the experimental realization, the gain medium consisting in 6 quantum wells embedded between a bottom totally reflective Bragg mirror and a top partially reflective Bragg mirror (½VCSEL) was considered; the ½VCSEL was then placed in an external cavity that was closed by a fast semiconductor saturable absorber mirror (SESAM) to operate the laser in the passive mode-locked regime (see fig. 1). In the absence of the optical feedback such a laser can generate so-called harmonic mode-locked solutions denoted by HML n where n is the number of equally spaced pulses per round-trip. The optical feedback induces that each pulse is followed by a train of small copies of itself. The size and position of this echo can be controlled via the feedback rate and the delay time, respectively. When an echo is placed close to the leading edge of another pulse, the pulse experiences less amplification by the gain medium as it is already depleted by the echo. This interaction can lead to the destruction of the main pulse. In consequence, the system settles on a solution where pulses and echos are well-separated and thus, do not interact. This mechanism is visualized in fig. 2 where a HML 4 and HML 2 are exposed to optical feedback and therefor, the output changes to a HML 3 and fundamental mode-locked solution, respectively. This behaviour was also observed in the experiment. In the further steps, a detailed bifurcation analysis is conducted where the influence of the position of the echo is investigated.

Passively mode-locked semiconductor VECSELs realized in a V-shaped cavity configuration ( Fig. 1a ) have proven to provide strong performance figures and are thus promising candidates for photonic applications. Furthermore, these lasers can also be a source of interesting pulsed dynamics, as the double gain pass within one round-trip in the external cavity favors the emergence of pulse clusters over harmonic mode-locking [1] . Here, we investigate this cavity effect in the regime of temporally localized states where the gain relaxation time has to be much shorter than the cavity length.

Spatial and temporal localized structures in optical cavities, also called Dissipative Solitons, have attracted a great deal of attention in the last two decades [1] , [2] . Beyond their fundamental interest, these structures can be used to code and process information bits in optical resonators. In general, two conditions appear to be necessary for a system to exhibit LSs: i) to have a large aspect-ratio and ii) to feature a multistable response. We have recently shown that an optically pumped passively mode-locked Vertical External Cavity Surface Emitting Laser (VECSEL) can be operated in a regime where mode-locked pulses becomes localized, i.e. they are individually addressable [3] , [4] . This regime requires long external cavity, i.e. cavity roundtrip time be significantly larger than gain recovery time, τ c >> τ g . Both the gain section and the saturable absorber mirrors needs to be properly engineered to provide multistability between different PML solutions with different number of pulses per roundtrip. Moreover, VECSEL geometry is very interesting because the external cavity can be designed in a self-imaging configuration which allow for reaching high-Fresnel number, hence paving the way also to spatial LSs [5] .

Passive-mode locking (PML) is a fascinating phenomenon allowing for the generation of ultrashort optical pulses. Semiconductor lasers are widely used as PML sources and Vertical External-Cavity Surface-Emitting Semiconductor Lasers (VECSELs) are explored in various applications that require stable lasing with high output power and good beam quality. Here, PML can be obtained by closing the external cavity with a semiconductor saturable absorber (SA) mirror (see Fig. 1 (a) ). Operating in the long cavity regime, where the round-trip time is much longer than the semiconductor gain recovery time, the PML pulses become independent and addressable temporal localized states (TLSs) below the lasing threshold [1] (see Fig. 1 (b) ).

We analyze the emergence of wiggling temporal localized states in a passively mode-locked Vertical External- Cavity Surface-Emitting Laser composed by a gain chip and a resonant saturable absorber mirror. We show that the wiggling instability stems from the interplay between the third-order dispersion induced by the micro-cavities and their frequency mismatch. The latter is identified as an experimentally crucial parameter defining the range of existence of stable emission. We reveal the homoclinic scenario underlying the wiggling phenomenon and we show how it allows to control the oscillation., 4 pages 5 figures, submitted to Optics Letters

In this paper, we analyze the effect of optical feedback on the dynamics of a passively mode-locked ring laser operating in the regime of temporal localized structures. This laser system is modeled by a system of delay differential equations, which include delay terms associated with the laser cavity and the feedback loop. Using a combination of direct numerical simulations and path-continuation techniques, we show that the feedback loop creates echos of the main pulse whose position and size strongly depend on the feedback parameters. We demonstrate that in the long-cavity regime, these echos can successively replace the main pulses, which defines their lifetime. This pulse instability mechanism originates from a global bifurcation of the saddle-node infinite-period type. In addition, we show that, under the influence of noise, the stable pulses exhibit forms of behavior characteristic of excitable systems. Furthermore, for the harmonic solutions consisting of multiple equispaced pulses per round-trip we show that if the location of the pulses coincide with the echo of another, the range of stability of these solutions is increased. Finally, it is shown that around these resonances, branches of different solutions are connected by period doubling bifurcations., Comment: 9 pages, 11 figures, published in "Chaos: An Interdisciplinary Journal of Nonlinear Science"

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. The analysis is carried out for the model in two spatial dimensions. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Passively mode-locked vertical external-cavity surface-emitting semiconductor lasers (VECSELs) composed of a gain chip and a semiconductor saturable absorber have been drawing much attention due to their excellent performance figures. In this work we investigate how localized structures and incoherent, nonlocally bound, pulse molecules emerge in a long cavity VECSEL using a $\mathsf{V}$-shaped cavity geometry. We show that these states are bistable with the laser off state and that they are individually addressable. Using a model based upon delay differential equations, we demonstrate that pulse clusters result from the cavity geometry and from the nonlocal coupling with the gain medium; in the long cavity regime this leads to locally independent, yet globally bound, phase incoherent photonic molecules. Using a multiple timescale analysis, we derive an amplitude equation for the field that allows us to predict analytically the distance between the elements of a cluster.

Time-delayed dynamical systems materialize in situations where distant, point-wise, nonlinear nodes exchange information that propagates at a finite speed. However, they are considered devoid of dispersive effects, which are known to play a leading role in pattern formation and wave dynamics. We show how dispersion may appear naturally in delayed systems and we exemplify our result by studying theoretically and experimentally the influence of third order dispersion in a system composed of coupled optical micro-cavities. Dispersion induced pulse satellites emerge asymmetrically and destabilize the mode-locking regime.

The time-delayed Adler equation is arguably the simplest model for an injected semiconductor laser with coherent injection and optical feedback. It is able to reproduce the existence of topological localized structures (LSs) and their rich interactions. In this paper we perform the first extended bifurcation analysis of this model and we explore the mechanisms by which LSs emerge. We also derive the effective equations governing the motion of distant LSs and we stress how the lack of parity in time-delayed systems leads to exotic, non-reciprocal, interactions between topological localized states., 13 pages, 14 figures, submitted to Chaos special issue on Pattern and solitons

The well-known cubic Allen–Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occurrence of different types of moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of the external field strength or chemical potential. We then employ the same methodology to systematically analyze fronts for more involved AC-type models. In particular, we consider a cubic–quintic variational AC model and two different nonvariational generalizations. We determine and compare the bifurcation diagrams of front solutions in the four considered models.

Spatially Localized States are individually addressable structures that may appear in large aspect-ratio optical resonators. They can be used as bits of information for all-optical buffering. We design and operate a modeless laser cavity based on a 1/2 VCSEL coupled to a distant mirror in self-imaging condition. Our study indicates how a VeCSEL can be specially designed to provide a robust system potentially capable of emitting Spatially Localized States and paves the way towards the observation of three dimensional - in space and time - confined states, the so-called light bullets.

The active Phase-Field-Crystal (aPFC) model combines elements of the Toner-Tu theory for self-propelled particles and the classical Phase-Field-Crystal (PFC) model that describes the transition between liquid to crystalline phases. In the liquid-crystal coexistence region of the PFC model, crystalline clusters exist in the form of localized states that coexist with the homogeneous background. At sufficiently strong self-propulsion strength (or activity) they start to travel. We employ numerical path continuation and direct time simulations to first investigate the existence regions of different types of localized states for a one-dimensional system. The results are summarized in morphological phase diagrams in the parameter plane spanned by activity and mean concentration. Then we focus on the interaction of traveling localized states studying their collision behavior. As a result we distinguish 'elastic' and 'inelastic' collisions. In the former, localized states recover their properties after a collision while in the latter they may annihilate or form resting or traveling bound states. In passing, we describe oscillating and modulated traveling localized states that have as the steadily traveling localized states no counterpart in the classical PFC model.

The passive conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) corresponds to a gradient dynamics for a single order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we first briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC equations described by common gradient dynamics, (ii) two coupled passive PFC where the coupling breaks the gradient dynamics structure, and (iii) a passive PFC coupled to an active PFC., Comment: submitted to the IMA Journal of Applied Mathematics' Special Issue on Homoclinic Snaking at 21, in memory of Patrick Woods

In this paper, we analyze the dynamics and formation mechanisms of bound states (BSs) of light bullets in the output of a laser coupled to a distant saturable absorber. First we approximate the full three-dimensional set of Haus master equations by a reduced equation governing the dynamics of the transverse profile. This effective theory allows us to perform a detailed multiparameter bifurcation study and to identify the different mechanisms of instability of BSs. In addition, our analysis reveals a non-intuitive dependence of the stability region as a function of the linewidth enhancement factors and the field diffusion. Our results are confirmed by direct numerical simulations of the full system., Comment: 8 pages 11 figures, submitted to Chaos special issue on nonequilibrium dynamics and patterns

We analyze the dynamics of passively mode-locked integrated external-cavity surface-emitting Lasers (MIXSELs) using a first-principle dynamical model based upon delay algebraic equations. We show that the third order dispersion stemming from the lasing micro-cavity induces a train of decaying satellites on the leading edge of the pulse. Due to the nonlinear interaction with carriers, these satellites may get amplified thereby destabilizing the mode-locked states. In the long cavity regime, the localized structures that exist below the lasing threshold are found to be deeply affected by this instability. As it originates from a global bifurcation of the saddle-node infinite period type, we explain why the pulses exhibit behaviors characteristic of excitable systems. Using the multiple time-scale and the functional mapping methods, we derive rigorously a master equation for MIXSELs in which third order dispersion is an essential ingredient. We compare the bifurcation diagram of both models and assess their good agreement., Comment: 14 pages, 11 figures, submitted to APS

When a solid substrate is withdrawn from a bath of simple, partially wetting, nonvolatile liquid, one typically distinguishes two regimes, namely, after withdrawal the substrate is macroscopically dry or homogeneously coated by a liquid film. In the latter case, the coating is called a Landau-Levich film. Its thickness depends on the angle and velocity of substrate withdrawal. We predict by means of a numerical and analytical investigation of a hydrodynamic thin-film model the existence of a third regime. It consists of the deposition of a regular pattern of liquid ridges oriented parallel to the meniscus. We establish that the mechanism of the underlying meniscus instability originates from competing film dewetting and Landau-Levich film deposition. Our analysis combines a marginal stability analysis, numerical time simulations and a numerical bifurcation study via path-continuation.

We consider the formation of temporal localized structures or Kerr-comb generation in a microresonator with inhomogeneities. We show that the introduction of even a small inhomogeneity in the injected beam widens the stability region of localized solutions. The homoclinic snaking bifurcation associated with the formation of localized structures and clusters of them with decaying oscillatory tails is constructed. Furthermore, the inhomogeneity allows us to not only control the position of localized solutions, but strongly affects their stability domains. In particular, a new stability domain of a single peak localized structure appears outside of the region of multistability between multiple peaks of localized states. We identify a regime of larger detuning, where localized structures do not exhibit a snaking behavior. In this regime, the effect of inhomogeneities on localized solutions is far more complex: They can act either attracting or repelling. We identify the pitchfork bifurcation responsible for this transition. Finally, we use a potential well approach to determine the force exerted by the inhomogeneity and summarize with a full analysis of the parameter regime, where localized structures and therefore Kerr-comb generation exist, and analyze how this regime changes in the presence of an inhomogeneity., SCOPUS: ar.j, info:eu-repo/semantics/published

Time-delayed dynamical systems materialize in situations where distant, pointwise, nonlinear nodes exchange information that propagates at a finite speed. However, they are considered devoid of dispersive effects, which are known to play a leading role in pattern formation and wave dynamics. We show how dispersion may appear naturally in delayed systems and we exemplify our result by studying theoretically and experimentally the influence of third order dispersion in a system composed of coupled optical microcavities. Dispersion-induced pulse satellites emerge asymmetrically and destabilize the mode-locking regime.

In this work we show that a high finesse Fabry-Perot micro-cavity containing a Kerr medium, coupled to a external cavity and in presence of optical injection can give rise to phase-locked temporal dissipative solitons. These Localized Structures (LSs) may possess a strongly asymmetrical oscillating tail which is the result of Third Order Dispersion induced by the cavity. Dispersion is found to induce soliton molecules and tunable Frequency Combs. Because of the external locking to the injection field, the LSs are phase coherent with each other, and could be used for interferometric applications.

Delayed dynamical systems (DDSs) describe a large number of phenomena in nature and they exhibit a wealth of dynamical regimes such as localized structures (LSs) [1], fronts and chimera states. A fertile perspective lies in their interpretation as spatially extended diffusive systems which holds in the limit of long delays [2]. A strong limitation of DDSs for this aim comes from the difficulty of taking into consideration chromatic dispersion. As such, DDSs description of wave propagation is limited to low dispersive media and/or narrow-spectrum signals. Nevertheless, chromatic dispersion plays a leading role in many phenomena occurring during wave evolution [3]. Beyond second order dispersion (SOD), third order dispersion (TOD) is the lowest order non-trivial parity symmetry breaking effect, which leads to convective instabilities [4] and drifts.

The possibility of using light bullets (LBs), i.e. pulses simultaneously confined in the transverse and the propagation directions, has attracted a lot of interest in the last twenty years, both for fundamental and practical reasons. In this work, we review the recent theoretical, numerical, and experimental advances regarding the realization of LBs in semiconductor passively mode-locked lasers. It was recently shown [1] that phase invariant Temporal Localized Structures (LSs) can evolve from passive mode-locking (PML). The PML regime leads to the emission of temporal pulses much shorter than the cavity round-trip and it is achieved by combining two elements, a laser amplifier providing a gain and a nonlinear loss element, usually a saturable absorber (SA). The different dynamical properties of the SA and of the gain create a window for regeneration only around the pulse. For cavities with a large aspect ratio, as defined by the ratio of the gain recovery time and of the cavity length r = τ/ τ g , the PML pulses may become individually addressable and coexist with the off solution [1]. It was later established in which conditions the transverse profile of these temporal LSs can self-organize, see Fig. 1, yielding a new robust LB formation scenario [2]. We present an intuitive theory [2,3] that allows to understand the LBs as hybrids between the temporal localized structures and spatial diffractive autosolitons. Using the path following techniques within pde2path framework, we performed the bifurcation analysis of the Haus master equation. We were able to unveil a new mechanism of instability for the both temporal [4], and spatial LSs which also appears as an instability for the full three-dimensional problem.

The influence of a temporal forcing on the pattern formation in Langmuir-Blodgett transfer is studied employing a generalized Cahn-Hilliard model. The occurring frequency locking effects allow for controlling the pattern formation process. In the case of one-dimensional (i.e., stripe) patterns one finds various synchronization phenomena such as entrainment between the distance of deposited stripes and the forcing frequency. In two dimensions, the temporal forcing gives rise to the formation of intricate complex patterns such as vertical stripes, oblique stripes and lattice structures. Remarkably, it is possible to influence the system in the spatial direction perpendicular to the forcing direction leading to synchronization in two spatial dimensions., Added sentences in reference to remarks from editors and fixed numbering, formatting issues

Soliton explosions are one of the most striking phenomena observed in mode-locked lasers. They were first predicted theoretically in a one-dimensional complex cubic-quintic Ginzburg-Landau equation (CQGLE) [1] in the case of anomalous linear dispersion and then were verified experimentally in a Kerr lens mode-locked Ti:sapphire laser [2], In this regime, a localized pulse circulating in the cavity experiences an abrupt structural collapse at certain points of the time evolution subsequently recovering its original shape. The experimentally observed explosions are similar but not identical to each other and the times between them appear to be randomly distributed. Among other reported features is the existence of symmetric and asymmetric explosions in a wide range of system parameters. Exploding solitary pulses have been recently observed in an all-normal-dispersion Yb-doped mode-locked fiber laser operated in a transition regime between stable and noise-like emission [3] and the experimental results have been successfully compared with realistic numerical simulations based on a model of complex pulse propagation in optical fibers. In [4], the connection of this model with the CQGLE with additional higher-order nonlinear and dispersive effects (HOEs) was established and the existence of periodic one-side explosions was reported. However, despite significant theoretical interest (see e.g. [5]), the impact of HOEs on the dynamics of a localized pulses and on the onset of soliton explosions have not been extensively studied so far.

A cavity soliton (CS) consists of one or more regions in one state surrounded by a region of a qualitatively different state. Such patterns may be stationary or oscillatory, static or moving and they are important in a wide variety of fields [1, 2]. They are potentially interesting for all-optical control of light. To this end, a delayed feedback can be used to control these CSs [3, 4]. Indeed, slowly moving spots subject to delay appear in various other fields. For example, spreading depolarization waves responsible for both migraine and strokes have been described as reaction-diffusion waves subject to delay [5]. Of particular interest are time delayed feedback control (TDEC) schemes. TDFC is based on the use of the difference between systems variables at the current moment of time and their values at some time in the past. They are smooth controls because the steady states of the unperturbed problem are solutions of the delayed feedback problem. More recently, we have investigated the combined influence of an inhomogeneity of the pumping and delayed feedback and shown that it induces oscillations and drift of localized structures in the Swift-Hohenberg equation describing pattern formation in the transverse plane of an optical cavity [6].

In this paper we study the transition to synchrony in an one-dimensional array of oscillators with non-local coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding Ott–Antonsen equation, which is an integro-differential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it was reported that in the spatially extended case at the synchronisation threshold there appear partially coherent plane waves with different wave numbers, which are organised in the well-known Eckhaus scenario. In this paper, we show that for Kuramoto–Sakaguchi phase oscillators the phase lag parameter in the interaction function can induce a Benjamin–Feir-type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves. We give an analytic treatment of the Benjamin–Feir instability and its onset in a codimension-two bifurcation in the Ott–Antonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the Benjamin–Feir unstable region.

We present a brief comparative investigation of the bifurcation structure related to the formation of two-dimensional deposition patterns as described by continuum models of Cahn–Hilliard type. These are, on the one hand a driven Cahn–Hilliard model for Langmuir–Blodgett transfer of a surfactant layer from the surface of a bath onto a moving plate and on the other hand a driven thin-film equation modelling the surface acoustic wave-driven coating of a plate by a simple liquid. In both cases, we present selected two-dimensional steady states corresponding to deposition patterns and discuss the main structure of the corresponding bifurcation diagrams.

In this paper, we analyze the formation and dynamical properties of discrete light bullets in an array of passively mode-locked lasers coupled via evanescent fields in a ring geometry. Using a generic model based upon a system of nearest-neighbor coupled Haus master equations, we show numerically the existence of discrete light bullets for different coupling strengths. In order to reduce the complexity of the analysis, we approximate the full problem by a reduced set of discrete equations governing the dynamics of the transverse profile of the discrete light bullets. This effective theory allows us to perform a detailed bifurcation analysis via path-continuation methods. In particular, we show the existence of multistable branches of discrete localized states, corresponding to different number of active elements in the array. These branches are either independent of each other or are organized into a snaking bifurcation diagram where the width of the discrete localized states grows via a process of successive increase and decrease of the gain. Mechanisms are revealed by which the snaking branches can be created and destroyed as a second parameter, e.g., the linewidth enhancement factor or the coupling strength is varied. For increasing couplings, the existence of moving bright and dark discrete localized states is also demonstrated.

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn–Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift–Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto–Sivashinsky equation, convective Allen–Cahn and Cahn–Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.

Recently, a mechanism of formation of light bullets (LBs) in wide-aperture passively mode-locked lasers was proposed. The conditions for existence and stability of these bullets, found in the long cavity limit, were studied theoretically under the mean field (MF) approximation using a Haus-type model equation. In this paper, we relax the MF approximation and study LB formation in a model of a wide-aperture three section laser with a long diffractive section and short absorber and gain sections. To this end, we derive a non-local delay-differential equation (NDDE) model and demonstrate by means of numerical simulations that this model supports stable LBs. We observe that the predictions about the regions of existence and stability of the LBs made previously using MF laser models agree well with the results obtained using the NDDE model. Moreover, we demonstrate that the general conclusions based upon the Haus model that regard the robustness of the LBs remain true in the NDDE model valid beyond the MF approximation, when the gain, losses and diffraction per cavity round trip are not small perturbations anymore. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.

We study the emergence and the stability of temporal localized structures in the output of a semiconductor laser passively mode-locked by a saturable absorber in the long cavity regime. For large yet realistic values of the linewidth enhancement factor, we disclose the existence of secondary dynamical instabilities where the pulses develop regular and subsequent irregular temporal oscil- lations. By a detailed bifurcation analysis we show that additional solution branches that consist in multi-pulse (molecules) solutions exist. We demonstrate that the various solution curves for the single and multi-peak pulses can splice and intersect each other via transcritical bifurcations, lead- ing to a complex web of solution. Our analysis is based upon a generic model of mode-locking that consists in a time-delayed dynamical system, but also upon a much more numerically efficient, yet approximate, partial differential equation. We compare the results of the bifurcation analysis of both models in order to assess up to which point the two approaches are equivalent. We conclude our analysis by the study of the influence of group velocity dispersion, that is only possible in the framework of the partial differential equation model, and we show that it may have a profound impact on the dynamics of the localized states.

The conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles, Menzel and Lowen [Phys. Rev. Lett. 110, 055702 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.055702] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.

We study the transverse profile of three-dimensional light bullets found theoretically in passively mode-locked laser. Using numerical path-continuation and time simulations, we discuss the range of existence and stability of these structures.

We present a novel approach for the analysis of passively mode-locked semiconductor lasers that allows for efficient parameter sweeps and time jitter analysis. It permits accessing the ultra-low repetition rate regime where pulses become localized states. The analysis including slow (e.g. thermal) processes or transverse dynamics becomes feasible. Our method bridges the divide between the phenomenological, yet highly efficient, pulse iterative model that is the Haus master equation, and the more involved first principle descriptions relying on time delayed equations. Our iterative functional mapping exploits the fundamental division of the mode-locking regime between fast and slow stages and allows computing the dynamics only in the pulse vicinity. Reductions of the simulation times and of the memory footprint up to two orders of magnitudes are demonstrated. Finally, the mapping also provides a general framework for deducing the Haus master equation from first principle models based upon delayed differential equations., Comment: Revised paper for Optics Letters

Vertical-external-cavity surface-emitting semi-conductor lasers (VECSELs) are of high interest for passive mode-locking (PML). They consist of coupled cavities that have unique filtering properties. We demonstrate that these can have profound implications on the pulse shape causing a new kind of instability for the pulse train. We focus on a MIXSEL system here for simplicity and clarity, however we stress that we found the same phenomena for a closely related VCSEL-RSAM system. Our setup is depicted in Fig.1 (upper) where we show the VCSEL intracavity field E and the field circulating in the external cavity Y. Following the approach in [1] we obtain a dynamical model linking both fields.

We investigate the spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subjected to time-delayed optical feedback. Using a combination of analytical, numerical, and path continuation methods, we analyze the bifurcation structure of stationary and moving cavity solitons and identify two different types of traveling localized solutions, corresponding to slow and fast motion. We show that the delay impacts both stationary and moving solutions either causing drifting and wiggling dynamics of initially stationary cavity solitons or leading to stabilization of intrinsically moving solutions. Finally, we demonstrate that the fast cavity solitons can be associated with a lateral mode-locking regime in a broad-area laser with a single longitudinal mode.

We briefly review selected mathematical models that describe the dynamics of pattern formation phenomena in dip-coating and Langmuir-Blodgett transfer experiments, where solutions or suspensions are transferred onto a substrate producing patterned deposit layers with structure length from hundreds of nanometres to tens of micrometres. The models are presented with a focus on their gradient dynamics formulations that clearly shows how the dynamics is governed by particular free energy functionals and facilitates the comparison of the models. In particular, we include a discussion of models based on long-wave hydrodynamics as well as of more phenomenological models that focus on the pattern formation processes in such systems. The models and their relations are elucidated and examples of resulting patterns are discussed before we conclude with a discussion of implications of the gradient dynamics formulation and of some related open issues.

Recently, the existence of robust three-dimensional light bullets (LBs) was predicted theoretically in the output of a laser coupled to a distant saturable absorber. In this paper, we analyze the stability and the range of existence of these dissipative localized structures and provide guidelines and realistic parameter sets for their experimental observation. In order to reduce the complexity of the analysis, we first approximate the three-dimensional problem by a reduced equation governing the dynamics of the transverse profile. This effective theory provides an intuitive picture of the LB formation mechanism. Moreover, it allows us to perform a detailed multiparameter bifurcation study and to identify the different mechanisms of instability. It is found that the LBs experience dominantly either homogeneous oscillation or symmetry-breaking transversal wave radiation. In addition, our analysis reveals several nonintuitive scaling behaviors as functions of the linewidth enhancement factors and the saturation parameters. Our results are confirmed by direct numerical simulations of the full system.

The wetting behavior of a liquid on solid substrates is governed by the nature of the effective interaction between the liquid-gas and the solid-liquid interfaces, which is described by the binding or wetting potential $g(h)$ which is an excess free energy per unit area that depends on the liquid film height $h$. Given a microscopic theory for the liquid, to determine $g(h)$ one must calculate the free energy for liquid films of any given value of $h$; i.e. one needs to create and analyze out-of-equilibrium states, since at equilibrium there is a unique value of $h$, specified by the temperature and chemical potential of the surrounding gas. Here we introduce a Nudged Elastic Band (NEB) approach to calculate $g(h)$ and illustrate the method by applying it in conjunction with a microscopic lattice density functional theory for the liquid. We show too that the NEB results are identical to those obtained with an established method based on using a fictitious additional potential to stabilize the non-equilibrium states. The advantages of the NEB approach are discussed., 5 pages, 2 figures